## Iterating transducers

### Dennis Dams, Yassine Lakhnech and Martin Steffen

To appear at 13th Conference on Computer-Aided Verification (CAV01), Paris, France, 18-22 July 2001

## Abstract

Regular languages have proved useful for the symbolic state exploration of infinite state systems. They can be used to represent infinite sets of system configurations; the transitional semantics of the system consequently can be modeled by finite-state transducers. A standard problem encountered when doing symbolic state exploration for infinite state systems is how to explore all states in a finite amount of time. When representing the one-step transition relation of a system by a finite-state transducer $\T$, this problem boils down to finding an appropriate finite-state representation $\Tmany$ for its transitive closure. In this paper we give a partial algorithm to compute a finite-state transducer $\Tmany$ for a general class of transducers, including transducers that are not length-preserving. The construction builds a quotient of an underlying infinite-state transducer $\Tmanyinfinite$, using a novel behavioural equivalence that is based \emph{past} and \emph{future} bisimulations computed on finite approximations of $\Tmanyinfinite$. The extrapolation to $\Tmanyinfinite$ of these finite bisimulations capitalizes on the structure of the states of $\Tmanyinfinite$, which are strings of states of $\T$. We show how this extrapolation may be rephrased as a problem of detecting \emph{confluence} properties of rewrite systems that represent the bisimulations. Thus, we can draw upon techniques that have been developed in the area of rewriting. As in general, $\Tmanyinfinite$ is not representable by a finite-state transducer, the construction may not terminate. Nevertheless our implementation is able to calculate the transitive closure on examples investigated by related symbolic techniques in the literature.

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